1. Fabric Blend Problem: State the problem: We want to find the amounts of cotton, polyester, and nylon in a fabric blend costing 3.25 per pound. Cotton costs 4.00, polyester 3.00, nylon 2.00 per pound. Nylon amount equals polyester amount. Let $x$ = pounds of cotton, $y$ = pounds of polyester, $z$ = pounds of nylon. Given $z = y$. The cost equation per pound of blend: $$\frac{4x + 3y + 2z}{x + y + z} = 3.25$$ Substitute $z = y$: $$\frac{4x + 3y + 2y}{x + y + y} = 3.25 \Rightarrow \frac{4x + 5y}{x + 2y} = 3.25$$ Multiply both sides: $$4x + 5y = 3.25(x + 2y) = 3.25x + 6.5y$$ Bring terms to one side: $$4x - 3.25x + 5y - 6.5y = 0 \Rightarrow 0.75x - 1.5y = 0$$ Simplify: $$0.75x = 1.5y \Rightarrow x = 2y$$ So cotton is twice polyester, nylon equals polyester. Final ratio: $$x : y : z = 2y : y : y = 2 : 1 : 1$$ 2. Taxes Problem: State the problem: Find federal and state taxes given taxable income 312000, federal tax is 25% of income after state tax, state tax is 10% of income after federal tax. Let $F$ = federal tax, $S$ = state tax. Federal tax: $$F = 0.25(312000 - S)$$ State tax: $$S = 0.10(312000 - F)$$ Substitute $S$ into $F$: $$F = 0.25(312000 - 0.10(312000 - F)) = 0.25(312000 - 31200 + 0.10F) = 0.25(280800 + 0.10F) = 70200 + 0.025F$$ Bring terms together: $$F - 0.025F = 70200 \Rightarrow 0.975F = 70200 \Rightarrow F = \frac{70200}{0.975} = 72000$$ Find $S$: $$S = 0.10(312000 - 72000) = 0.10(240000) = 24000$$ 3. Airplane Speed Problem: State the problem: Plane travels 900 mi in 3 h with tailwind, return trip 3 h 36 min (3.6 h) against wind. Find plane speed in still air $p$ and wind speed $w$. With wind: $$p + w = \frac{900}{3} = 300$$ Against wind: $$p - w = \frac{900}{3.6} = 250$$ Add equations: $$2p = 300 + 250 = 550 \Rightarrow p = 275$$ Subtract equations: $$2w = 300 - 250 = 50 \Rightarrow w = 25$$ Final answers: Cotton: 2 parts, Polyester: 1 part, Nylon: 1 part. Federal tax: 72000. State tax: 24000. Plane speed: 275 mph. Wind speed: 25 mph.